11.1 First-order and continuous phase transitions

Phase Transitions

Phase transitions describe how a system shifts from one thermodynamic phase to another. Understanding the distinction between first-order and continuous transitions is central to predicting how materials behave near transition points, from the latent heat released when water freezes to the diverging fluctuations near a ferromagnet's Curie temperature.

First-order vs continuous phase transitions

These two categories are distinguished by how the Gibbs free energy and its derivatives behave at the transition point.

First-order phase transitions involve a discontinuous jump in the first derivatives of the Gibbs free energy (entropy $S$ and volume $V$). At the transition temperature, two distinct phases coexist. You can literally watch ice and liquid water sitting together at 0°C, each with a different density and entropy. Other examples include liquid-gas boiling and solid-solid allotropic transformations (like graphite converting to diamond under pressure).

Continuous (second-order) phase transitions have no such jump. The first derivatives of the free energy change smoothly through the transition. Instead, it's the second derivatives (heat capacity, compressibility, susceptibility) that diverge at the critical point. The transition happens without phase coexistence and without latent heat. Classic examples include:

• The ferromagnetic-paramagnetic transition at the Curie temperature, where spontaneous magnetization smoothly drops to zero
• The superconducting transition at the critical temperature $T_c$
• The superfluid transition in helium-4 at the lambda point (~2.17 K)

Discontinuities in thermodynamic quantities

For first-order transitions, the Gibbs free energy $G$ itself stays continuous across the transition (both phases have the same $G$ at the transition point, which is the equilibrium condition). The discontinuities show up in its first derivatives:

• Entropy: There's a finite jump $\Delta S$ at the transition temperature. This entropy discontinuity is directly tied to latent heat through $L = T\,\Delta S$.
• Volume: There's a finite jump $\Delta V$. The two coexisting phases have different densities. Think of how ice is less dense than liquid water, which is why ice floats.

Because $G$ is continuous but its first derivatives are not, you can picture the $G(T)$ curve as having a kink (a change in slope) at the transition temperature. The slope of $G$ with respect to $T$ gives $-S$, so a kink in $G$ means a jump in $S$.

Latent heat in phase transitions

Latent heat is the energy a substance absorbs or releases during a first-order phase transition while its temperature stays constant. All of that energy goes into rearranging the molecular structure (breaking bonds in melting, separating molecules in vaporization) rather than raising the temperature.

The relationship is:

$L = T\,\Delta S$

where $T$ is the transition temperature and $\Delta S$ is the entropy difference between the two phases.

Two common types:

• Latent heat of fusion: Energy to convert solid to liquid. For water, this is about 334 J/g at 0°C.
• Latent heat of vaporization: Energy to convert liquid to gas. For water, this is about 2260 J/g at 100°C. Vaporization requires much more energy because you're fully separating molecules against their intermolecular attractions.

Continuous transitions, by contrast, have $\Delta S = 0$ at the critical point, so there is no latent heat.

Continuity in second-order transitions

In a continuous transition, the first derivatives of the free energy (entropy, volume) change smoothly. There's no abrupt jump and no latent heat. What does happen is more subtle and, in many ways, more interesting.

The order parameter is the quantity that distinguishes the two phases. For a ferromagnet, it's the magnetization $M$. As temperature approaches the critical temperature $T_c$ from below, $M$ continuously decreases and reaches zero exactly at $T_c$. Above $T_c$, the system is paramagnetic.

Near $T_c$, several quantities exhibit power-law divergences characterized by critical exponents:

• Heat capacity: $C \propto |T - T_c|^{-\alpha}$
• Order parameter (below $T_c$): $M \propto (T_c - T)^{\beta}$
• Susceptibility: $\chi \propto |T - T_c|^{-\gamma}$

The correlation length $\xi$, which measures the distance over which microscopic fluctuations are correlated, also diverges as $T \to T_c$. This divergence is why the system becomes scale-invariant at the critical point, and it's the physical basis for universality: systems with very different microscopic physics (fluids, magnets, superfluids) share the same critical exponents if they belong to the same universality class.